The loop integral of Eq. (REF ) in redshift space can be
similarly evaluated in redshift space. The difference between real
space and redshift space is that the second-order propagator is given
by Eq. (REF ) in the latter, instead of Eq. (REF ) in
the former. The product of tripolar spherical harmonics reduces to a
single tripolar spherical harmonics according to a formula
[1]}, [2]}
\(\left\lbrace Y_{l_{z1}}(\hat{{z}}) \otimes \left\lbrace Y_{l_1^{\prime }}(\hat{{k}}^{\prime }) \otimes Y_{l_1^{\prime \prime }}(\hat{{k}}^{\prime \prime })\right\rbrace _{L_1}\right\rbrace _{l_1m_1}\\ \times \left\lbrace Y_{l_{z2}}(\hat{{z}})\otimes \left\lbrace Y_{l_2^{\prime }}(\hat{{k}}^{\prime }) \otimes Y_{l_2^{\prime \prime }}(\hat{{k}}^{\prime \prime })\right\rbrace _{L_2}\right\rbrace _{l_2m_2}\\= \frac{(-1)^{l_1+l_2}}{(4\pi )^{3/2}}\sqrt{\lbrace l_1\rbrace \lbrace l_2\rbrace  \lbrace l_{z1}\rbrace \lbrace l_{z2}\rbrace \lbrace l_1^{\prime }\rbrace \lbrace l_2^{\prime }\rbrace  \lbrace l_1^{\prime \prime }\rbrace \lbrace l_2^{\prime \prime }\rbrace \lbrace L_1\rbrace \lbrace L_2\rbrace }\\ \times \sum _{l} \sqrt{\lbrace l\rbrace }\left(l_1\,l_2\,l\right)_{m_1m_2}^{\phantom{m_1m_2}m}\sum _{l_z,l^{\prime },l^{\prime \prime },L} (-1)^{l_z+l^{\prime }+l^{\prime \prime }}\sqrt{\lbrace l_z\rbrace \lbrace l^{\prime }\rbrace \lbrace l^{\prime \prime }\rbrace \lbrace L\rbrace }\\ \times \begin{pmatrix}l_{z1} & l_{z2} & l_z \\0 & 0 & 0\end{pmatrix}\begin{pmatrix}l_1^{\prime } & l_2^{\prime } & l^{\prime } \\0 & 0 & 0\end{pmatrix}\begin{pmatrix}l_1^{\prime \prime } & l_2^{\prime \prime } & l^{\prime \prime } \\0 & 0 & 0\end{pmatrix}\begin{Bmatrix}l_1 & l_2 & l \\l_{z1} & l_{z2} & l_z \\L_1 & L_2 & L\end{Bmatrix}\\ \times \begin{Bmatrix}L_1 & L_2 & L \\l_1^{\prime } & l_2^{\prime } & l^{\prime } \\l_1^{\prime \prime } & l_2^{\prime \prime } & l^{\prime \prime }\end{Bmatrix}\left\lbrace Y_{l_z}(\hat{{z}}) \otimes \left\lbrace Y_{l^{\prime }}(\hat{{k}}^{\prime }) \otimes Y_{l^{\prime \prime }}(\hat{{k}}^{\prime \prime })\right\rbrace _L\right\rbrace _{lm}.\) 
